Approach of the value of a rent when non-central moments of the capitalization factor are known: an R application with interest rates following normal and beta distributions Ratio Mathematica Volume 45, 2023 Pell Even Sum Cordial Labeling of Graphs Christina Mercy A* Tamizh Chelvam T† Abstract Let𝐺 = (𝑉, 𝐸) be a simple graph and let 𝑃𝑖 be Pell numbers. For a bijection𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃|𝑉|βˆ’1}, assign the label 1 for the edge 𝑒 = 𝑒𝑣 if 𝑓(𝑒) + 𝑓(𝑣) is even and label 0 otherwise. Then 𝑓 is said to be a Pell even sum cordial labeling of 𝐺 if |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (1)| ≀ 1 where 𝑒𝑓 (0) and 𝑒𝑓 (1) denote the number of edges labeled with 0 and 1 respectively. If any graph admits Pell even sum cordial labeling, it is called Pell even sum cordial graph. In this study, we show that star, comb, bistar, jewel, crown, bipartite graph πΎπ‘š,π‘š, flower graph, helm, wheel, triangular book, 𝐾2 + π‘šπΎ1 are Pell even sum cordial. Keywords: Cordial Labeling, Pell Numbers, Pell Even Sum Cordial Labeling. 2010 AMS Subject Classification: 05C78‑. *Research Scholar, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627 012, Tamilnadu, India. e-mail: mercyudhayan@gmail.com. †CSIR Emeritus Scientist, Department of Mathematics, Manonmaniam Sundaranar University, Tirunelveli-627 012, Tamilnadu, India. ‑ Received on July 10, 2022. Accepted on October 15, 2022. Published on January 30, 2023. doi: 10.23755/rm. v45i0.1034. ISSN: 1592-7415. eISSN: 2282-8214. Β©The Authors. This paper is published under the CC-BY license agreement. 304 Christina Mercy A and Tamizh Chelvam T 1. Introduction By a graph 𝐺 = (𝑉, 𝐸) we mean a finite, undirected simple graph. We refer to Harary [5] for graph theory concepts and refer Gallian [4] for the literature on graph labeling. The Pell numbers are defined by the recurrence relation 𝑃𝑛 = 2π‘ƒπ‘›βˆ’1 + π‘ƒπ‘›βˆ’2 where 𝑃0 = 0 and 𝑃1 = 1. The Pell number sequence is given as 0,1,2,5,12, …Cahit [2] is credited for inventing cordial labeling. Shiama [11] defined Pell labeling and shown that path, cycle, star, double star, coconut tree, bistar and π΅π‘š,𝑛,π‘˜ are Pell graphs. Muthu Ramakrishnan and Sutha[8] proposed Pell graceful labeling as an extension of Fibonacci graceful labeling and demonstrated that cycle, path, olive tree, comb graph are Pell graceful graphs whereas complete graph and wheel graphs are not Pell graceful. Indira et al. [6] suggested some algorithms for the existence of Pell labeling in quadrilateral snake, extended duplicate graph of quadrilateral graph. Sriram et al. [12] investigated the Pell labeling for the joins of square of a path. Muthu Ramakrishnan and Sutha [7] also demonstrated that bistar, subdivision of bistar, caterpillar graphs, Jelly fish graph, star graph, coconut tree are pell graceful. Sharon Philomena and Thirusangu [10] shown that < 𝐾1,𝑛 ∢ 2 > is a Pell graph. Avudainayaki and Selvam [1] shown that the extended duplicate graph of arrow graph and splitting graph of path admits harmonious and Pell labeling. Celine Mary et al. [3] demonstrated through an algorithm that inflation of alternate triangular snake graph of odd length in which the alternate block starts from the second vertex is a Pell graph. Muthu Ramakrishnan and Sutha [9] proposed the Pell square graceful labeling and proved that subdivision of the edges of a path𝑃𝑛 in 𝑃𝑛 βŠ™ 𝐾1, < 𝑆𝑛: π‘š >, Olive tree, twig graph are Pell square graceful. Inspired by the concepts discussed above, the Pell even sum cordial labeling is being introduced here. It is defined as follows. Definition 1.1 Let𝐺 = (𝑉, 𝐸) be a simple graph and let 𝑃𝑖 be Pell numbers. For a bijection𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃|𝑉|βˆ’1}, assign the label 1 for the edge 𝑒 = 𝑒𝑣 if 𝑓(𝑒) + 𝑓(𝑣) is even and label 0 otherwise. Then 𝑓 is said to be a Pell even sum cordial labeling of 𝐺 if |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (1)| ≀ 1 where 𝑒𝑓 (0) and 𝑒𝑓 (1) denote the number of edges labeled with 0 and 1 respectively. If any graph admits Pell even sum cordial labeling, it is called Pell even sum cordial graph. Definition 1.2 The Comb 𝑃𝑛 βŠ™ 𝐾1 is the graph created by adding a pendent edge to each vertex of a path. Definition 1.3 The bistar 𝐡𝑛,𝑛 is the graph obtained by joining the apex vertices of two copies of 𝐾1,𝑛. Definition 1.4 The jewel graph 𝐽𝑛 is a graph with vertex set 𝑉(𝐽𝑛 ) = {𝑒, π‘₯, 𝑣, 𝑦, 𝑣𝑖 : 1 ≀ 𝑖 ≀ 𝑛} and the edge set 𝐸(𝐽𝑛) = {𝑒π‘₯, 𝑣π‘₯, 𝑒𝑦, 𝑣𝑦, π‘₯𝑦, 𝑒𝑣𝑖 , 𝑣𝑣𝑖 : 1 ≀ 𝑖 ≀ 𝑛}. 305 Pell Even Sum Cordial Labeling of Graphs Definition 1.5 The crown 𝐢𝑛 βŠ™ 𝐾1 is the graph obtained from a cycle by attaching a pendent edge to each vertex of the cycle. Definition 1.6 A complete bipartite is a graph whose vertices can be partitioned into two subsets 𝑉1 and 𝑉2 such that no edge has both endpoints in the same subset, and every possible edge that could connect vertices in different subsets is part of the graph. Definition 1.7 The helm is the graph obtained from a wheel graph by adjoining a pendent edge at each node of the cycle. Definition 1.8 The Flower 𝐹𝑙𝑛 is the graph obtained from a helm by attaching each pendent vertex to the apex of the helm. Definition 1.9 The join of two graphs 𝐺1 and 𝐺2 is denoted by 𝐺1 + 𝐺2 and whose vertex set is 𝑉(𝐺1 + 𝐺2) = 𝑉(𝐺1) βˆͺ 𝑉(𝐺2) and edge set 𝐸(𝐺1 + 𝐺2) = 𝐸(𝐺1) βˆͺ 𝐸(𝐺2) βˆͺ {𝑒𝑣 ∢ 𝑒 ∈ 𝐺1, 𝑣 ∈ 𝐺2}. Definition 1.10 One edge union of cycles of same length is called a book. The common edge is called the base of the book. If we consider 𝑑 copies of cycles of length 𝑛 β‰₯ 3, then the book is denoted by 𝐡𝑛 𝑑 . If 𝑛 = 3,4,5 or 6, the book 𝐡 is called book with triangular, rectangular, pentagonal or hexagonal pages respectively. Definition 1.11 The Wheel π‘Šπ‘› can be defined as the graph join 𝐾1 + πΆπ‘›βˆ’1. Definition 1.12 𝐾1,kis a tree with one internal node and π‘˜ leaves. 2. Pell Even Sum Cordial Graphs In this section, we prove that star, comb, bistar, jewel, crown, bipartite graph 𝐾𝑛,𝑛, flower graph, helm, wheel, triangular book, 𝐾2 + π‘šπΎ1 are Pell even sum cordial. Theorem 2.1. For 𝑛 β‰₯ 2, the comb is Pell even sum cordial graph Proof. Recall that comb 𝐺 = (𝑉, 𝐸) is a graph obtained from the path with vertices 𝑒1, 𝑒2, … , 𝑒𝑛 by joining a vertex 𝑣𝑖 to each 𝑒𝑖 where 1 ≀ 𝑖 ≀ 𝑛. Actually |𝑉(𝐺)| = 2𝑛 and |𝐸(𝐺)| = 2𝑛 βˆ’ 1. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2π‘›βˆ’1} defined by 𝑓(𝑒𝑖 ) = 𝑃2π‘–βˆ’1 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2π‘–βˆ’2 for 1 ≀ 𝑖 ≀ 𝑛 . The induced edge labels are given 𝑓 βˆ—(𝑒𝑖 𝑒𝑖+1) = 1 for 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 and 𝑓 βˆ—(𝑒𝑖 𝑣𝑖 ) = 0 for 1 ≀ 𝑖 ≀ 𝑛. From this, 𝑒𝑓 (0) = 𝑛, 𝑒𝑓 (1) = 𝑛 βˆ’ 1 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for 𝑛 β‰₯ 2, the comb is a Pell even sum cordial graph. Theorem 2.2.For 𝑛 β‰₯ 2, the bistar π΅π‘š,π‘š is a Pell even sum cordial. Proof. Consider the bistar 𝐺 = π΅π‘š,π‘š. Then 𝑉(𝐺) = {𝑒, 𝑣, 𝑒𝑖 , 𝑣𝑖 : 1 ≀ 𝑖 ≀ π‘š}, 𝐸(𝐺) = {𝑒𝑣, 𝑒𝑒𝑖 , 𝑣𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ π‘š}, |𝑉(𝐺)| = 2π‘š + 2 and |𝐸(𝐺)| = 2π‘š + 1. Consider 306 Christina Mercy A and Tamizh Chelvam T 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2π‘š+1} defined by 𝑓(𝑒) = 𝑃0, 𝑓(𝑣) = 𝑃1, 𝑓(𝑒𝑖 ) = 𝑃𝑖+1 for 1 ≀ 𝑖 ≀ π‘š βˆ’ 1 and 𝑓(𝑣𝑖 ) = π‘ƒπ‘š+𝑖 for 1 ≀ 𝑖 ≀ π‘š. The induced edge labels are given by 𝑓 βˆ—(𝑒𝑣) = 0. For 1 ≀ 𝑖 ≀ π‘š βˆ’ 1, 𝑓 βˆ—(𝑒𝑒𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). For 1 ≀ 𝑖 ≀ π‘š, 𝑓 βˆ—(𝑣𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). When π‘š is odd, 𝑒𝑓 (1) = π‘š + 1, 𝑒𝑓 (0) = π‘š and on the otherhand, when π‘š is even, 𝑒𝑓 (1) = π‘š , 𝑒𝑓 (0) = π‘š + 1. Therefore, for 𝑛 β‰₯ 2, bistar π΅π‘š,π‘š is Pell even sum cordial. Theorem 2.3. For 𝑛 β‰₯ 1, the jewel graph 𝐽𝑛 is Pell even sum cordial. Proof. Consider the jewel graph 𝐺 = 𝐽𝑛 . Here 𝑉(𝐺) = {𝑒, 𝑣, π‘₯, 𝑦, 𝑒𝑖 : 1 ≀ 𝑖 ≀ 𝑛}, 𝐸(𝐺) = {𝑒π‘₯, 𝑒𝑦, π‘₯𝑦, π‘₯𝑣, 𝑦𝑣, 𝑒𝑒𝑖 , 𝑣𝑣𝑖 : 1 ≀ 𝑖 ≀ 𝑛}, |𝑉(𝐺)| = 𝑛 + 4 and |𝐸(𝐺)| = 2𝑛 + 5. Let 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃𝑛+3} be defined by 𝑓(𝑒) = 𝑃0, 𝑓(𝑣) = 𝑃1, 𝑓(π‘₯) = 𝑃2 and 𝑓(𝑒𝑖 ) = 𝑃𝑖+3 for 1 ≀ 𝑖 ≀ 𝑛. Then the induced edge labels are given by,𝑓 βˆ—(𝑒𝑣) = 1, 𝑓 βˆ—(𝑒𝑦) = 1, 𝑓 βˆ—(π‘₯𝑦) = 0, 𝑓 βˆ—(𝑣π‘₯) = 0 and 𝑓 βˆ—(𝑣𝑦) = 1 For 1 ≀ 𝑖 ≀ 𝑛, 𝑓 βˆ—(𝑒𝑒𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2). 𝑓 βˆ—(𝑣𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). From the above, 𝑒𝑓 (0) = 𝑛 + 3, 𝑒𝑓 (1) = 𝑛 + 2 and so|𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for 𝑛 β‰₯ 1, the jewel graph 𝐽𝑛 is Pell even sum cordial. Theorem 2.4 For 𝑛 β‰₯ 3, the crown 𝐢𝑛 βŠ™ 𝐾1 is Pell even sum cordial. Proof. Consider the crown 𝐺 = 𝐢𝑛 βŠ™ 𝐾1. Here 𝑉(𝐺) = {𝑒𝑖 , 𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛} and 𝐸(𝐺) = {𝑒𝑖 𝑒𝑖+1: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1; 𝑒𝑛𝑒1, 𝑒𝑖 𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛}, |𝑉(𝐺)| = 2𝑛 = |𝐸(𝐺)|. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2π‘›βˆ’1} defined by 𝑓(𝑒𝑖 ) = 𝑃2π‘–βˆ’2 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2π‘–βˆ’1 for 1 ≀ 𝑖 ≀ 𝑛. Then the induced edge labels are given by 𝑓 βˆ—(𝑒𝑖 𝑒𝑖+1) = 1 for 1 ≀ 𝑖 ≀ 𝑛, 𝑓 βˆ—(𝑒𝑛𝑒1) = 1 and 𝑓 βˆ—(𝑒𝑖 𝑣𝑖 ) = 0 for 1 ≀ 𝑖 ≀ 𝑛. From the above, 𝑒𝑓 (0) = 𝑛 + 3, 𝑒𝑓 (1) = 𝑛 + 2 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for 𝑛 β‰₯ 1, the crown 𝐢𝑛 βŠ™ 𝐾1 is Pell even sum cordial. Theorem 2.5 For 𝑛 β‰₯ 2, the complete bipartite graph 𝐾𝑛,𝑛 is Pell even sum cordial. Proof. Let 𝐺 = 𝐾𝑛,𝑛. Let the partitions of vertex set be 𝑉1 = {𝑒1, 𝑒2, … , 𝑒𝑛 }and 𝑉2 = {𝑣1, 𝑣2, … , 𝑣𝑛 }. Hence 𝑉(𝐺) = {𝑒1, 𝑒2, … , 𝑒𝑛 , 𝑣1, 𝑣2, … , 𝑣𝑛 }and 𝐸(𝐺) = {𝑒𝑖 π‘£π’Š ∢ 1 ≀ 𝑖 ≀ 𝑛}. Then |𝑉(𝐺)| = 2𝑛 , |𝐸(𝐺)| = π‘›πŸ. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2π‘›βˆ’1}defined by 𝑓(𝑒𝑖 ) = π‘ƒπ‘–βˆ’1for 1 ≀ 𝑖 ≀ 𝑛 and𝑓(𝑣𝑖 ) = 𝑃(π‘›βˆ’1)+𝑖 for 1 ≀ 𝑖 ≀ 𝑛. The induced edge labels are given by, 307 Pell Even Sum Cordial Labeling of Graphs For 1 ≀ 𝑖 ≀ 𝑛, 𝑓 βˆ—(𝑒𝑖 𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1,0 (π‘šπ‘œπ‘‘ 2); 0 π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. When 𝑛 is odd, 𝑒𝑓 (0) = 𝑛2+1 2 , 𝑒𝑓 (1) = 𝑛2βˆ’1 2 and when, 𝑛 is even, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 𝑛2 2 . Therefore, for 𝑛 β‰₯ 2, the complete bipartite graph 𝐾𝑛,𝑛 is Pell even sum cordial. Theorem 2.6 For 𝑛 β‰₯ 3, the flower graph 𝐹𝑙𝑛 is Pell even sum cordial. Proof. Let 𝐺 = 𝐹𝑙𝑛. Then 𝑉(𝐺) = {𝑣, 𝑒𝑖 , 𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛} and 𝐸(𝐺) = {𝑣𝑣𝑖 , 𝑣𝑖 𝑒𝑖 , 𝑣𝑒𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 ; 𝑣𝑛 𝑣1 ; 𝑣𝑖 𝑣𝑖+1 ∢ 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1}. Then,|𝑉(𝐺)| = 2𝑛 + 1, |𝐸(𝐺)| = 4𝑛. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … 𝑃2𝑛} defined by 𝑓(𝑣𝑛) = 𝑃2𝑛 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑣𝑖 ) = 𝑃2π‘–βˆ’2 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑒𝑖 ) = 𝑃2π‘–βˆ’1 for 1 ≀ 𝑖 ≀ 𝑛. The induced edge labels are given by 𝑓 βˆ—(𝑣𝑖 𝑣𝑖+1) = 1, 𝑓 βˆ—(𝑣𝑖 𝑒𝑖 ) = 0, 𝑓 βˆ—(𝑣𝑒𝑖 ) = 0, 𝑓 βˆ—(𝑣𝑣𝑖 ) = 1 and 𝑓 βˆ—(𝑣𝑛𝑣1) = 1 for 1 ≀ 𝑖 ≀ 𝑛. From the above, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 2𝑛 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, the flower graph 𝐹𝑙𝑛 is Pell even sum cordial for 𝑛 β‰₯ 3. Theorem 2.7 For 𝑛 β‰₯ 3 and 𝑛 is even, the Helm 𝐻𝑛 is Pell even sum cordial. Proof. Let 𝐺 = 𝐻𝑛. Then 𝑉(𝐺) = {𝑣, 𝑒𝑖 , 𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛}, 𝐸(𝐺) = {𝑒𝑖 𝑒𝑖+1: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 ; 𝑒𝑖 𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 ; 𝑣𝑒𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 ; 𝑒1𝑒𝑛}, |𝑉(𝐺)| = 2𝑛 + 1, |𝐸(𝐺)| = 3𝑛. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2𝑛 } defined by 𝑓(𝑒𝑖 ) = 𝑃𝑖 for 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1, 𝑓(𝑣𝑖 ) = 𝑃𝑛+𝑖 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑣) = 0. Then the induced edge labels are given by, For 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1, 𝑓 βˆ—(𝑒𝑖 𝑒𝑖+1) = 0, 𝑓 βˆ—(𝑣𝑖 𝑒𝑖 ) = 1, 𝑓 βˆ—(𝑒𝑛𝑒1) = 0 and 𝑓 βˆ—(𝑣𝑒𝑖 ) = { 0 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 1 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). From the above, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 3𝑛 2 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for an even integer β‰₯ 3 , the Helm 𝐻𝑛 is Pell even sum cordial. Theorem 2.8 For 𝑛 β‰₯ 4, the wheel graph π‘Šπ‘› is Pell even sum cordial. Proof. Let 𝐺 = π‘Šπ‘›. Then 𝑉(𝐺) = {𝑒0, 𝑒1, π‘’πŸ, … , 𝑒𝑛 } and (𝐺) = {𝑒𝑖 𝑒𝑖+1: 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1 ; 𝑒0𝑣𝑖 : 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1}, |𝑉(𝐺)| = 𝑛 + 1 and |𝐸(𝐺)| = 2𝑛 βˆ’ 2. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃2𝑛 } defined by 𝑓(𝑒𝑖 ) = 𝑃2π‘–βˆ’2 for 1 ≀ 𝑖 ≀ 𝑛 and 𝑓(𝑒0) = 𝑃1. Then the induced edge labels are given by 𝑓 βˆ—(𝑒𝑖 𝑒𝑖+1) = 1 for 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1, 𝑓 βˆ—(𝑒𝑛𝑒1) =1 and 𝑓 βˆ—(𝑒0𝑒𝑖 ) = 0 for 1 ≀ 𝑖 ≀ 𝑛. From this, 𝑒𝑓 (0) = 𝑒𝑓 (1) = 𝑛 βˆ’ 1 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for 𝑛 β‰₯ 4, the wheel graph π‘Šπ‘› is Pell even sum cordial. Theorem 2.9 For 𝑛 β‰₯ 2, the star graph 𝐾1,𝑛 is Pell even sum cordial. Proof. Let 𝐺 = 𝐾1,𝑛. Then 𝑉(𝐺) = {𝑣, 𝑒1, 𝑒2, … , 𝑒𝑛} and 𝐸(𝐺) = {𝑣𝑣𝑖 ∢ 1 ≀ 𝑖 ≀ 𝑛}, |𝑉(𝐺)| = 𝑛 + 1 and |𝐸(𝐺)| = 𝑛. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , 𝑃𝑛 } defined by 𝑓(𝑣) = 𝑃0, 𝑓(𝑒𝑖 ) = 𝑃𝑖 for 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 1. The induced edge labels are given by, for 1 ≀ 𝑖 ≀ 𝑛, 𝑓 βˆ—(𝑣𝑒𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2). 308 Christina Mercy A and Tamizh Chelvam T When 𝑛 is odd, 𝑒𝑓 (0) = n+1 2 , 𝑒𝑓 (1) = π‘›βˆ’1 2 whereas when 𝑛 is even,𝑒𝑓 (0) = n 2 = 𝑒𝑓 (1).Therefore, for 𝑛 β‰₯ 2, the star graph 𝐾1,𝑛 is Pell even sum cordial. Theorem 2.10 For 𝑛 β‰₯ 3 , the triangular book 𝑇𝐡𝑛 is Pell even sum cordial. Proof. Let 𝐺 = 𝑇𝐡𝑛 . Then 𝑉(𝐺) = {𝑣1, 𝑣2, … , π‘£π‘›βˆ’1, 𝑣𝑛}, 𝐸(𝐺) = {𝑣1𝑣𝑖 , 𝑣𝑛 𝑣𝑖 , 𝑣𝑛 𝑣1 ∢ 2 ≀ 𝑖 ≀ 𝑛 βˆ’ 1} , |𝑉(𝐺)| = 𝑛 and |𝐸(𝐺)| = 2𝑛 βˆ’ 3. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , π‘ƒπ‘›βˆ’1} defined by 𝑓(𝑣1) = 𝑃0 , 𝑓(𝑣𝑛 ) = 𝑃1 and 𝑓(𝑣𝑖 ) = 𝑃𝑖+1 for 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 2. Then the induced edge labels are given by 𝑓 βˆ—(𝑣1𝑣𝑛) = 0. For 1 ≀ 𝑖 ≀ 𝑛 βˆ’ 2, 𝑓 βˆ—(𝑣1𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2). 𝑓 βˆ—(𝑣𝑛 𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). From the above, 𝑒𝑓 (0) = 𝑛 βˆ’ 1, 𝑒𝑓 (1) = 𝑛 βˆ’ 2 and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for 𝑛 β‰₯ 3 , the triangular book 𝑇𝐡𝑛 is Pell even sum cordial. Theorem 2.11. For π‘š β‰₯ 3, the graph 𝐾2 + π‘šπΎ1 is Pell even sum cordial. Proof. Let 𝐺 = 𝐾2 + π‘šπΎ1. Then (𝐺) = {𝑒1, 𝑒2, 𝑣1, 𝑣2 … , π‘£π‘š} , 𝐸(𝐺) = {𝑒1𝑒2, 𝑒1𝑣𝑖 , 𝑒2𝑣𝑖 : 1 ≀ 𝑖 ≀ π‘š} , |𝑉(𝐺)| = π‘š + 2 and |𝐸(𝐺)| = 2π‘š + 1. Consider 𝑓: 𝑉(𝐺) β†’ {𝑃0, 𝑃1, … , π‘ƒπ‘š+1} defined by 𝑓(𝑒1) = 𝑃0, 𝑓(𝑒2) = 𝑃1 and 𝑓(𝑣𝑖 ) = 𝑃𝑖+1 for 1 ≀ 𝑖 ≀ π‘š. Then the induced edge labels are given by 𝑓 βˆ—(𝑒1𝑒2) = 0. For 1 ≀ 𝑖 ≀ π‘š, 𝑓 βˆ—(𝑒1𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2). 𝑓 βˆ—(𝑒2𝑣𝑖 ) = { 1 𝑖𝑓 𝑖 ≑ 0 (π‘šπ‘œπ‘‘ 2); 0 𝑖𝑓 𝑖 ≑ 1 (π‘šπ‘œπ‘‘ 2). From the above, 𝑒𝑓 (0) = π‘š + 1, 𝑒𝑓 (0) = π‘š and so |𝑒𝑓 (0) βˆ’ 𝑒𝑓 (0)| ≀ 1. Therefore, for π‘š β‰₯ 3, the graph 𝐾2 + π‘šπΎ1 is Pell even sum cordial. 3. 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